Practice Paper Pure 3 (Edexcel International A Level Maths: Pure 4)

Simplify fully

 

 

 

On the same axes sketch the graphs of  and  where 

Find an expression for  and state its domain.

Show that 

Scientists introduced a small number of rare breed deer to a large wildlife sanctuary.

The population of deer, within the sanctuary, is modelled by

 is the number of deer after  years of first being introduced to the sanctuary.

Write down the number of deer the scientists introduced to the sanctuary.

How many years does it take for the deer's population to double?

Give one criticism of the model for population growth.

The scientists suggest that the population of deer are separated after either 25 years or when their population exceeds 400.
Find the earliest time the deer should be separated.

By letting , use the identity for to derive an expression for in terms of tan .

Hence, or otherwise, solve the equation

Use calculus to show that

The diagram below shows part of the graph of  , whereis the function defined by

Points A, B and C are the three places where the graph intercepts the x-axis.

Find .

Show that the coordinates of point A are (-2, 0).

Find the equation of the tangent to the curve at point A.

Show that the equation

can be written as

Rewrite the equation in the form  .

Given that ,

Find    in terms of y

Hence find in terms of x.

According to legend, unicorn tears have magical healing powers.

Without unicorn tears, a bruise of initial size A mm² will heal according to the model

where f is the area of the bruise, in square millimetres, at time t days since the bruise first appeared.

Use the iterative formula

with to find how many days it takes a bruise to heal without unicorn tears. Give your answer to three significant figures.

With unicorn tears, a bruise of the same initial size will heal according to the model

where time T is measured in seconds.

Using your answers from parts (a) and (b), work out approximately how many times quicker a bruise heals when unicorn tears are used.

Show that

Hence, or otherwise, find an expression for